Section: New Results

Markovian models for tumor growth

Participants: T. Bastogne, R. Keinj, P. Vallois.

Our research in this direction includes two contributions in 2011:

• A multinomial model for cell growth allowing to calibrate radiotherapies given in [3] .

• A study of tumor growth based on the lifespan of each cell (see [13] ).

More specifically, our two contributions can be summarized as follows:

(i) Hit and target models of tumor growth typically assume that all surviving cells have a constant and homogeneous sensitivity during the radiotherapy period. In [3] , we propose a multinomial model based on a discrete-time Markov chain, able to take into account cell repair, cell damage heterogeneity and cell proliferation. The proposed model relies on the 'Hit paradigm' and 'Target' theory in radiobiology and assumes that a cancer cell contains $m$ targets which must be all deactivated to produce cell death. The surviving cell population is then split up into $m$ categories to introduce the variation of cancer cell radio-sensitivity according to their damage states. Two other parameters have been introduced : the probability $q$ for a target to be deactivated by radiation and the probability $r$ for an inactive target in an alive cell to be reactivated. The parameter $q$ is related to the radiation dose $u_0$ through the intrinsic sensitivity of a target to radiation. Moreover, the multinomial model is a generalization of typical hit models. Based on the multinomial model, new expressions of the TCP (Tumor Control Probability) and NTCP (Normal Tissue Complication Probability) have been proposed for nonuniform radiations which permits to deduce the optimal total dose to be delivered. We point out the important influence of the repair parameter $r$ which could lead to reduce both the total radiation dose to be delivered and the risk of side effects.

(ii) We have proposed in [13] an original approach that expresses the probability distribution of the cancer and normal cells lifespans in terms of the number of dose fractions in radiotherapy. Conversely to previous models that examines the number of surviving cells in the treated population at fixed time instants, our modeling approach better reveals the dynamics of the tumor response.

We start by considering the lifespan of a single cancer cell that behaves as described in [3] . We study this random time by calculating its mean, variance and cumulative distribution function. We then assume that a tumor is a group of independent cells. This allows to define the lifespan of the tumor as the maximum of individual lifespans. When the initial number $n_0$ of cancer cells is not too large, then we can explicitly calculate the mean, variance and the cumulative distribution function of the tumor lifespan. When $n_0$ is large, the previous parameters are no longer calculable. However, we are able to show that, under some assumptions, the mean lifespan of the tumor behaves as a logarithmic function of the initial number $n_0$. The second goal is to show that TCP and NTCP can be completely formulated with respect to the tumor and normal tissue lifespans. These expressions of TCP and NTCP are finally used to propose a ROC curve, called ECT (Efficiency-Complication Trade-off), suited to the determination of the appropriate treatment schedule. This synthetic representation summarizes both efficiency and complication of the treatment. Moreover, it allows several possibilities of choice for the radiotherapist : treatment efficiency, priority to safety of normal tissue, or a trade-off between them.